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Problem 7.3.35
22–36. Derivatives Find the derivatives of the following functions.
f(x) = x sinh⁻¹ x − √(x² + 1)
Problem 7.3.10
How does the graph of the catenary y = a cosh x/a change as a > 0 increases?
Problem 7.3.37
37–56. Integrals Evaluate each integral.
∫ cosh 2x dx
Problem 7.3.45
37–56. Integrals Evaluate each integral.
∫₀ ˡⁿ ² tanh x dx
Problem 7.3.39
37–56. Integrals Evaluate each integral.
∫ sinh x / (1 + cosh x) dx
Problem 7.3.41
37–56. Integrals Evaluate each integral.
∫ tanh²x dx (Hint: Use an identity.)
Problem 7.3.44
37–56. Integrals Evaluate each integral.
∫₀⁴ sech²√x / √x dx
Problem 7.3.47
37–56. Integrals Evaluate each integral.
∫ dx/(8 – x²), x > 2√2
Problem 7.3.42
37–56. Integrals Evaluate each integral.
∫ sinh²z dz (Hint: Use an identity.)
Problem 7.3.38
37–56. Integrals Evaluate each integral.
∫ sech² w tanh w dw
Problem 7.3.38b
37–38. Caffeine After an individual drinks a beverage containing caffeine, the amount of caffeine in the bloodstream can be modeled by an exponential decay function, with a half-life that depends on several factors, including age and body weight. For the sake of simplicity, assume the caffeine in the following drinks immediately enters the bloodstream upon consumption.
An individual consumes two cups of coffee, each containing 90 mg of caffeine, two hours apart. Assume the half-life of caffeine for this individual is 5.7 hours.
b. Determine the amount of caffeine in the bloodstream 1 hour after drinking the second cup of coffee.
Problem 7.3.43d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. If the rate constant of an exponential growth function is increased, its doubling time is decreased.
Problem 7.3.49
37–56. Integrals Evaluate each integral.
∫ eˣ/(36 – e²ˣ), x < ln 6
Problem 7.3.56
37–56. Integrals Evaluate each integral.
∫₂₅²²⁵ dx / (x² + 25x) (Hint: √(x² + 25x) = √x √(x + 25).)
Problem 7.3.57
57–58. Two ways
Evaluate the following integrals two ways.
a. Simplify the integrand first and then integrate.
b. Change variables (let u = ln x), integrate, and then simplify your answer. Verify that both methods give the same answer.
∫ (sinh (ln x)) / x dx
Problem 7.3.50
37–56. Integrals Evaluate each integral.
∫ dx/x√(16 + x²)
Problem 7.3.63
63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.
∫₁ᵉ^² dx/x√(ln²x + 1)
Problem 7.3.65
63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.
∫₋₂² dt/(t² – 9)
Problem 7.3.79a
Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.
a. cosh 0
Problem 7.3.79d
Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.
d. sech (sinh 0)
Problem 7.3.79g
Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.
g. cosh² 1
Problem 7.3.79j
Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.
j. sinh⁻¹ (e² − 1)/2e
Problem 7.3.78a
Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.
a. coth 4
Problem 7.3.78c
Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.
c. csch⁻¹ 5
Problem 7.3.78f
Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.
f. tan⁻¹(sinh x) |₋₃³
Problem 7.3.74a
Wave velocity Use Exercise 73 to do the following calculations.
a. Find the velocity of a wave where λ = 50 m and d = 20 m.
Problem 7.3.75a
Shallow-water velocity equation
a. Confirm that the linear approximation to ƒ(x) = tanh x at a = 0 is L(x) = x.
Problem 7.3.77c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. ln(1 + √2) = −ln(−1 + √2)
Problem 7.3.81
Critical points Find the critical points of the function ƒ(x) = sinh² x cosh x.
Problem 7.3.83
Points of inflection Find the x-coordinate of the point(s) of inflection of f(x) = tanh² x.